Finite volume methods

FMDenaro · September 15, 2023, 04:13

Quote:

Originally Posted by hunt_mat

1) Correct. The flux for the conservation of mass involves u only.
2) Essentially, the first cell is at dh/2.
3) There is a diffusion equation for the velocity. The negative velocity is diffused in at h=1.
4) Yes.

So, you confirm that x=0 is a face location, not a cell-centre location.
How do you compute (in both equations) the diffusive fluxes at x=0?

hunt_mat · September 15, 2023, 05:20

Quote:

Originally Posted by FMDenaro

So, you confirm that x=0 is a face location, not a cell-centre location.
How do you compute (in both equations) the diffusive fluxes at x=0?

The flux for the mass equation, the fluxes are just the velocity, and that's 0 at x=0. For the momentum, I use the same gradient from 0 to dx/2, as I do from dx/2 to 3dx/2.

FMDenaro · September 15, 2023, 05:53

Quote:

Originally Posted by hunt_mat

The flux for the mass equation, the fluxes are just the velocity, and that's 0 at x=0. For the momentum, I use the same gradient from 0 to dx/2, as I do from dx/2 to 3dx/2.

But at x=0 how is the formula for the diffusive flux?

hunt_mat · September 15, 2023, 06:04

Quote:

Originally Posted by FMDenaro

But at x=0 how is the formula for the diffusive flux?

I use the same value for the derivative at h=dh/2.

FMDenaro · September 15, 2023, 06:07

Quote:

Originally Posted by hunt_mat

I use the same value for the derivative at h=dh/2.

But you do not have the same computational stencil at x=0 since you have no longer nodes at x<0.

hunt_mat · September 18, 2023, 06:54

Quote:

Originally Posted by FMDenaro

But you do not have the same computational stencil at x=0 since you have no longer nodes at x<0.

At the first cell, I compute the derivative in the cell centre and I use that for the edge.

FMDenaro · September 18, 2023, 07:22

Quote:

Originally Posted by hunt_mat

At the first cell, I compute the derivative in the cell centre and I use that for the edge.

But how do you compute that? Do you introduce the bc value this way?
Expand explicitly the scheme in the first cell.

hunt_mat · September 18, 2023, 08:34

Quote:

Originally Posted by FMDenaro

But how do you compute that? Do you introduce the bc value this way?
Expand explicitly the scheme in the first cell.

I compute the derivative as:
$\frac{\partial u}{\partial h}\approx\frac{u_{2}-u_{1}}{\frac{1}{2}(\delta h_{1}+\delta h_{2})}$

FMDenaro · September 18, 2023, 09:20

Quote:

Originally Posted by hunt_mat

I compute the derivative as:
$\frac{\partial u}{\partial h}\approx\frac{u_{2}-u_{1}}{\frac{1}{2}(\delta h_{1}+\delta h_{2})}$

1 and 2 are the centers of the FV?
I am not sure what you did but if I assume that rhe node i=1 is at x=h/2, the face x=0 has a first order accurate derivative given by

du/dx=u1/(h/2)

since u=0 at x=0

hunt_mat · September 18, 2023, 10:17

Quote:

Originally Posted by FMDenaro

1 and 2 are the centers of the FV?
I am not sure what you did but if I assume that rhe node i=1 is at x=h/2, the face x=0 has a first order accurate derivative given by

du/dx=u1/(h/2)

since u=0 at x=0

The 1 and 2 are at the cell centres and the node at i=1 is at dh_1/2. The equation you provided does not solve the problem I mentioned.

FMDenaro · September 18, 2023, 11:04

Quote:

Originally Posted by hunt_mat

The 1 and 2 are at the cell centres and the node at i=1 is at dh_1/2. The equation you provided does not solve the problem I mentioned.

However, your previous formula is wrong for the flux at x=0.
What if you set to zero the diffusive flux at x=0? I suspect that u=0 as BC is not congruent. Physically you will have an increasing of the kinetic energy in the domain.

hunt_mat · September 18, 2023, 11:44

Quote:

Originally Posted by FMDenaro

However, your previous formula is wrong for the flux at x=0.
What if you set to zero the diffusive flux at x=0? I suspect that u=0 as BC is not congruent. Physically you will have an increasing of the kinetic energy in the domain.

So that means I have two boundary conditions at h=0. Won't this be unstable?

FMDenaro · September 18, 2023, 11:56

Quote:

Originally Posted by hunt_mat

So that means I have two boundary conditions at h=0. Won't this be unstable?

From a mathematical point, you have two equations, each one has its suitable BC depending on the mathematical classification.

From a physical point of view, I see that you let mass entering for the right side (u is negative), that means you have also kinetic energy (rho*u^2/2) increasing into the domain. If u=0 at the left side, there is no mass leaving the domain, right?

hunt_mat · September 18, 2023, 12:04

Quote:

Originally Posted by FMDenaro

From a mathematical point, you have two equations, each one has its suitable BC depending on the mathematical classification.

From a physical point of view, I see that you let mass entering for the right side (u is negative), that means you have also kinetic energy (rho*u^2/2) increasing into the domain. If u=0 at the left side, there is no mass leaving the domain, right?

Wrong in both cases. The mass of the system is constant. What I allow at h=1, is the stress free condition which allows the movement of the boundary.

hunt_mat · September 18, 2023, 12:25

Quote:

Originally Posted by FMDenaro

From a mathematical point, you have two equations, each one has its suitable BC depending on the mathematical classification.

From a physical point of view, I see that you let mass entering for the right side (u is negative), that means you have also kinetic energy (rho*u^2/2) increasing into the domain. If u=0 at the left side, there is no mass leaving the domain, right?

I put in zero flux at h=0, and I still get change.

FMDenaro · September 18, 2023, 12:29

Quote:

Originally Posted by hunt_mat

Wrong in both cases. The mass of the system is constant. What I allow at h=1, is the stress free condition which allows the movement of the boundary.

I never worked on this kind of problem, but that seems just like a piston moving in the negative direction in a closed system and compressing the flow. At a certain point compression should stop, you cannot have a constant mass in a volume going to zero.
What should be the difference from the classic gasdynamics problem for the simple wave case as shown in Zucrow?

FMDenaro · September 18, 2023, 12:31

Quote:

Originally Posted by hunt_mat

I put in zero flux at h=0, and I still get change.

Could you show the plots of your solution at several time instants?

LuckyTran · September 18, 2023, 16:09

Do you set gradients or fluxes to 0 at x=0?

du/dx=0 is setting the boundary/face gradient to 0 which will constrain both the advective and diffusive flux.

You optionally can set the convective and diffusive fluxes individually to zero. This is the better way to go for FVM since FVM calls explicitly for face fluxes and not face gradients. But to answer your previous question, setting diffusive fluxes to zero does not give you too many constraints, it should have been one of the original constraints in the problem definition.

hunt_mat · September 19, 2023, 06:32

Quote:

Originally Posted by LuckyTran

Do you set gradients or fluxes to 0 at x=0?

du/dx=0 is setting the boundary/face gradient to 0 which will constrain both the advective and diffusive flux.

You optionally can set the convective and diffusive fluxes individually to zero. This is the better way to go for FVM since FVM calls explicitly for face fluxes and not face gradients. But to answer your previous question, setting diffusive fluxes to zero does not give you too many constraints, it should have been one of the original constraints in the problem definition.

I'm using a Lagrangian formalism, so I have no advective diffusion. I set the flux at h=0 to be zero, but I still have the same issue.

LuckyTran · September 19, 2023, 08:53

Oh sorry, that's right you have only a diffusive flux in the lagrangian formulation. Now just keep in mind that you can set the flux to 0 without invoking any differencing scheme. This is the way.

The other way is to set the du/dx to zero and then interpolate it onto the face flux. This way introduces discretization errors and inconsistencies.

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September 18, 2023, 16:09		#158
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,735 Rep Power: 66	Do you set gradients or fluxes to 0 at x=0? du/dx=0 is setting the boundary/face gradient to 0 which will constrain both the advective and diffusive flux. You optionally can set the convective and diffusive fluxes individually to zero. This is the better way to go for FVM since FVM calls explicitly for face fluxes and not face gradients. But to answer your previous question, setting diffusive fluxes to zero does not give you too many constraints, it should have been one of the original constraints in the problem definition.

September 19, 2023, 08:53		#160
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,735 Rep Power: 66	Oh sorry, that's right you have only a diffusive flux in the lagrangian formulation. Now just keep in mind that you can set the flux to 0 without invoking any differencing scheme. This is the way. The other way is to set the du/dx to zero and then interpolate it onto the face flux. This way introduces discretization errors and inconsistencies.